2012 AIME I Problems/Problem 1
Problem 1
Find the number of positive integers with three not necessarily distinct digits, , with
and
such that both
and
are multiples of
.
Solution
A positive integer is divisible by if and only if its last two digits are divisible by 4. For any value of
, there are two possible values for
and
, since we find that if
is even,
and
must be either
or
, and if
is odd,
and
must be either
or
. Thus, there are
ways to choose
and
, and
ways to choose
(since
can be any digit). Therefore, the final answer is
.
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
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All AIME Problems and Solutions |